Using results of regression on raw observation values to approximate results of regression on relative change between observations (Simple, Linear)

Question

this is my first time on Stack Exchange so if I did something wrong please tell me.

I have a time series dataset.
There is an observation $(y,x)$ for each continuous time $t$.
Let’s say for each day since 2014-01-01, i.e. $t \in \left\{0, 1, 2, …, 108 \right\}$, where $108$ means today.

I ran a regression on the model
$y = \alpha + \beta x$
and found $\alpha$ and $\beta$ with $R^2 > 0.975$

My question is, can I approximate $\beta'$ in the model
$y' = \alpha' + \beta'x'$
where
$y'=\frac{y_{t+1}}{y_t}-1$
$x'=\frac{x_{t+1}}{x_t}-1$
without running regression on that model,

just by using $\hat{\beta'}= \frac{\beta \times x_{108}}{\hat{y_{108}}}$

where $\hat{y_{108}}$ is estimated using the original model.

I'm curious because I found that my $\hat{\beta'}$ is usually within 1% of $\beta'$

Even a simple Yes or No with help me a lot.

Thanks!

Answer 1

The answer is "it depends".

It depends on what is the true process which generated this data, aka DGP (data generation process). Your approximation will not hold in a general case.

I'll give you an example.

Let's assume the true DGP: $\ln y_t=\alpha+\beta \ln x_t+\varepsilon_t$, where $\varepsilon_t\sim\mathcal{N}(0,\sigma)$

Differenceing gives: $\Delta \ln y_{t+1}=\ln\frac{y_{t+1}}{y_{t}}=\beta\ln \frac{x_{t+1}}{x_t}+\Delta\varepsilon_{t+1}$

Since $\Delta \ln y_{t+1} \approx\frac{y_{t+1}}{y_{t}}-1=y_t'$, you get $y_t'=\beta x_t'+\Delta\varepsilon_{t}$

Now, if in your sample $x_t,y_t\sim 1$ then $\ln y_t\approx y_t-1$, , plugging this into DGP we get your model:

$y_t=\tilde{\alpha}+\beta x_t+\varepsilon_t$

So, my guess is that your $x_t,y_t$ happened to be close to 1, that's why it's working for you. Now that I wrote this all, I figured there's a simpler way to show this, but I'm too lazy to re-write it :)